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<p><dfn class="terminology">Remark</dfn> Consider the situation where <span class="process-math">\(r=\rho\)</span> is a triple-repeated eigenvalue of (<a href="" class="xref" data-knowl="./knowl/eq7_14.html" title="Equation 6.4.3">(6.4.3)</a>):<dfn class="terminology">Case i.</dfn> There is only one eigenvector <span class="process-math">\(\vec{\xi}^{(1)}\)</span> and one solution is</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq7_14.html ./knowl/eq7_13_1.html ./knowl/eq7_13_1.html">
\begin{equation*}
{\bf x}^{(1)}=\vec{\xi}^{(1)} e^{\rho t}.
\end{equation*}
</div>
<p class="continuation">The second and third solutions have the form of</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq7_14.html ./knowl/eq7_13_1.html ./knowl/eq7_13_1.html">
\begin{equation*}
{\bf x}^{(2)}=(t \vec{\xi}^{(2)}+\vec{\eta}^{(2)}) e^{\rho t},\quad
{\bf x}^{(3)}=(t^2 \vec{\xi}^{(3)}+t \vec{\eta}^{(3)}+\vec{\zeta}^{(3)})  e^{\rho t},
\end{equation*}
</div>
<p class="continuation">where <span class="process-math">\(\vec{\xi}^{(2)}, \vec{\eta}^{(2)},  \vec{\xi}^{(3)}, \vec{\eta}^{(3)}, \vec{\zeta}^{(3)}\)</span> can be determined from (<a href="" class="xref" data-knowl="./knowl/eq7_13_1.html" title="Equation 6.4.1">(6.4.1)</a>).<dfn class="terminology">Case ii</dfn> There are two linear independent eigenvectors <span class="process-math">\(\vec{\xi}^{(1)}\)</span> and <span class="process-math">\(\vec{\xi}^{(2)}\text{.}\)</span> Two solutions are</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq7_14.html ./knowl/eq7_13_1.html ./knowl/eq7_13_1.html">
\begin{equation*}
{\bf x}^{(1)}=\vec{\xi}^{(1)} e^{\rho t},\quad
{\bf x}^{(2)}=\vec{\xi}^{(2)} e^{\rho t}.
\end{equation*}
</div>
<p class="continuation">The third solution has the form</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq7_14.html ./knowl/eq7_13_1.html ./knowl/eq7_13_1.html">
\begin{equation*}
{\bf x}^{(3)}=(t \vec{\xi}^{(3)}+\vec{\eta}^{(3)}) e^{\rho t}
\end{equation*}
</div>
<p class="continuation">and by taking it into the (<a href="" class="xref" data-knowl="./knowl/eq7_13_1.html" title="Equation 6.4.1">(6.4.1)</a>), we have</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq7_14.html ./knowl/eq7_13_1.html ./knowl/eq7_13_1.html">
\begin{equation}
({\bf A}-\rho {\bf I}) \vec{\xi}^{(3)}=0,\quad
({\bf A}-\rho {\bf I}) \vec{\eta}^{(3)}=\vec{\xi}^{(3)},\tag{6.4.9}
\end{equation}
</div>
<p class="continuation">where <span class="process-math">\(\vec{\xi}^{(3)}\)</span> should be one eigenvector corresponding to <span class="process-math">\(r=\rho\text{.}\)</span> Most generally, it should be</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq7_14.html ./knowl/eq7_13_1.html ./knowl/eq7_13_1.html">
\begin{equation*}
\vec{\xi}^{(3)}=k_1 \vec{\xi}^{(1)}+k_2 \vec{\xi}^{(2)}
\end{equation*}
</div>
<p class="continuation">where we should choose such values for <span class="process-math">\(k_1\)</span> and <span class="process-math">\(k_2\)</span> that <span class="process-math">\((\ref{eq7_16})_2\)</span> has a solution for <span class="process-math">\(\vec{\eta}^{(3)}\text{.}\)</span></p>
<span class="incontext"><a href="sec6_4.html#p-277" class="internal">in-context</a></span>
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